If vector $\vec{a} = 2\hat{i} - 3\hat{j} + 6\hat{k}$ and vector $\vec{b} = -2\hat{i} + 2\hat{j} - \hat{k},$ then $\frac{\text{Projection of vector } \vec{a} \text{ on vector } \vec{b}}{\text{Projection of vector } \vec{b} \text{ on vector } \vec{a}} = $

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of the vertices $A, B, C$ of a triangle respectively. Find the area of the triangle.

The moment of the force $\overrightarrow{F}$ acting at a point $P$,about the point $C$ is:

If $a, b$ and $c$ are unit vectors such that $a + b - c = 0,$ then the angle between $a$ and $b$ is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that $\vec{b} \times \vec{a}=\vec{c} \times \vec{a}$. If the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$,then the greatest integer less than or equal to $\tan^2 \theta$ is:

If vectors satisfy the condition $|a - c| = |b - c|$,then $(b - a) \cdot \left( c - \frac{a + b}{2} \right)$ is equal to